qml.ops.sk_decomposition

sk_decomposition(op, epsilon, *, max_depth=5, basis_set=('T', 'T*', 'H'), basis_length=10)

Approximate an arbitrary single-qubit gate in the Clifford+T basis using the Solovay-Kitaev algorithm.

This method implements the Solovay-Kitaev decomposition algorithm that approximates any single-qubit operation with \(\epsilon > 0\) error. The procedure exits when the approximation error becomes less than \(\epsilon\), or when max_depth approximation passes have been made. In the latter case, the approximation error could be \(\geq \epsilon\).

This algorithm produces a decomposition with \(O(\text{log}^{3.97}(1/\epsilon))\) operations.

Parameters

• op (Operation) – A single-qubit gate operation.

• epsilon (float) – The maximum permissible error.

Keyword Arguments

• max_depth (int) – The maximum number of approximation passes. A smaller \(\epsilon\) would generally require a greater number of passes. Default is 5.

• basis_set (list[str]) – Basis set to be used for the decomposition and building an approximate set internally. It accepts the following gate terms: ['X', 'Y', 'Z', 'H', 'T', 'T*', 'S', 'S*'], where * refers to the gate adjoint. Default value is ['T', 'T*', 'H'].

• basis_length (int) – Maximum expansion length of Clifford+T sequences in the internally-built approximate set. Default is 10.

Returns

A list of gates in the Clifford+T basis set that approximates the given operation along with a final global phase operation. The operations are in the circuit-order.

Return type

list[Operation]

Raises

ValueError – If the given operator acts on more than one wires.

Example

Suppose one would like to decompose RZ with rotation angle \(\phi = \pi/3\):

import numpy as np
import pennylane as qml

op  = qml.RZ(np.pi/3, wires=0)

# Get the gate decomposition in ['T', 'T*', 'H']
ops = qml.ops.sk_decomposition(op, epsilon=1e-3)

# Get the approximate matrix from the ops
matrix_sk = qml.prod(*reversed(ops)).matrix()

When the function is run for a sufficient depth with a good enough approximate set, the output gate sequence should implement the same operation approximately.

>>> qml.math.allclose(op.matrix(), matrix_sk, atol=1e-3)
True

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